Question

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{1}{x-2}\) \([1,4]\)

Short answer

The intervals of continuity for the function \(f(x)=\frac{1}{x-2}\) within the given range [1,4] are \((1, 2)\) and \((2, 4)\).

Step-by-step solution

Identify the Domain of the Function

The function \(f(x)=\frac{1}{x-2}\) is undefined at x = 2. This is because when x = 2, the denominator of the function is 0. So the domain of the function is all real numbers excluding 2: \((-\infty, 2) \cup (2, \infty)\)

Sketch the Graph of the Function

Sketch the graph of the function while keeping in mind that there's a vertical asymptote at x = 2. This means that the function approaches but never reaches that point. Draw two curves on either side of x = 2, showing the infinite nature of the function near that point.

Identify the Intervals of Continuity

The function is continuous wherever it is defined. Hence, its intervals of continuity fall within its domain. Considering the given range from 1 to 4, the function is continuous in two intervals: from 1 to 2 and from 2 to 4. But we must remember to exclude 2 from these intervals because the function is not defined at that point. Thus, the intervals of continuity within the given range are \((1, 2)\) and \((2, 4)\).

Key concepts

Domain of a Function

Understanding the domain of a function is an important first step in analyzing how a function behaves. The domain is essentially all possible input values (usually denoted by "x") for a given function. For the function \(f(x)=\frac{1}{x-2}\), you find the domain by identifying any values of \(x\) that would make the function undefined.
\(\frac{1}{x-2}\) becomes undefined when its denominator is zero, which occurs when \(x=2\). Therefore, \(x=2\) cannot be part of the domain.

• The domain of \(f(x)\) is all real numbers except 2.
• This can be expressed as: \((-\infty, 2) \cup (2, \infty)\).

By excluding 2, we ensure that our function remains defined for every \(x\) in the domain. Understanding the domain helps when sketching the graph, plotting only the valid input values. It also tells you how the function will behave at specific values.

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never touches. It's most commonly seen in rational functions, such as \(f(x)=\frac{1}{x-2}\).
In this function, the vertical asymptote occurs at \(x=2\). But why is that? Because as \(x\) approaches 2, the denominator \(x-2\) approaches zero, making the function value go to infinity or negative infinity.

• The vertical asymptote is at \(x=2\).
• This means the graph will get infinitely close to the line \(x=2\) but never actually touch or cross it.

When sketching a graph, vertical asymptotes indicate a "limit" behavior. The curve will seem to "shoot" upwards or downwards as it approaches the vertical asymptote.

Interval of Continuity

The interval of continuity of a function is where the function is unbroken and uninterrupted. A function is continuous on an interval if you can draw its graph over that interval without lifting your pencil from the paper. For the function \(f(x)=\frac{1}{x-2}\), it is continuous wherever it is defined, which means anywhere in its domain excluding any undefined points.
This function is continuous on the intervals \((1, 2)\) and \((2, 4)\) when considering the given range \([1, 4]\).

• Between \(x=1\) and \(x=2\), the function is continuous but not inclusive of \(x=2\).
• Between \(x=2\) and \(x=4\), similarly, it's continuous except at \(x=2\).

Understanding the intervals of continuity is crucial, as it tells you where the function behaves predictably, without jumps, holes, or interruptions.